Curvature flows with local and non-local sources

• Applicant: Professor Dr. Julian Scheuer
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 548204661

This projects deals with the motion of hypersurfaces driven by extrinsic curvature and forcing terms, with the additional property that certain geometric quantities are constant along the evolution. In previous works on the subject, the forcing terms either constituted a globally defined term, such as a certain integral quantity, or a local term, for example one that depends on the position of the surface. The novelty of this project is that we consider mixed forcing terms, so that they have global and local elements. This provides more flexibility in the flow design, which enabled us (Esther Cabezas-Rivas and Julian Scheuer) to solve a long-standing open problem in the field, namely the deformation of convex hypersurfaces of the unit sphere by a modified mean curvature flow to a round limit, while an arbitrary quermassintegral is preserved. This project to supposed to improve and extend this first result into several directions, for example towards the deformation of free boundary hypersurfaces.

• DFG Programme: Research Grants
• International Connection: Spain
• Cooperation Partner: Professorin Dr. Esther Cabezas-Rivas